1. Exploratory Spatial Optimization in Site Search: A Neighborhood Operator Approach Thomas J. Cova Department of Geography University of Utah and Richard L. Church Department of Geography University of California Santa Barbara

2. Outline Site search problems GIS and spatial optimization approaches Integrating site search models with GIS A neighborhood operator approach Explicit contiguity constraints (1-D and 2-D) Exploratory spatial optimization in site search Conclusions and future research

3. Site search problems Locate and configure a finite area (site) optimally to serve a given land use or activity.  A site’s suitability, area, cost and spatial relationships with other geographic features are important considerations.  Spatial site characteristics like shape, contiguity, and enclaves are also important considerations.  There are no candidate sites from which to select a best site. Applications: Siting a residential subdivision, hazardous waste site, landfill, biological reserve, recreational site, or other areal feature.

4. Representing a Site continuous space raster modelirregular polygons

5. GIS Approaches to Site Search Suitability mapping Score each land unit according to its ability to support a given land use (McHarg, 1969; Hopkins, 1977; Lyle and Stutz, 1983; Burrough, 1997). Land screening Eliminate land units from consideration as elements of a site based on their attributes (Dobson, 1978). Suitability map high low Land screening Not screened screened

6. GIS II: Iterative Relaxation Lower a high-water plane until an area sufficient to meet the area siting requirements appears. suitability Advantages Simple. Computationally friendly (Eastman et al. 1993). Drawbacks Little to no control over site area, shape, and location. Limited ability to trade one characteristic against another.

7. Spatial Optimization Approaches Land acquisition, land allocation, site allocation, and site search models of Wright et al. (1983), Gilbert et al. (1985), Diamond & Wright (1991), Minor & Jacobs (1994), Williams & Revelle (1996), and Brookes (1997). A site is an aggregation of individual spatial units.

8. The Contiguity Problem 1. Encourage contiguity with a model (Wright, et al., 1983; Minor & Jacobs, 1994; Williams & Revelle, 1996). How can we guarantee a contiguous solution to a site search problem? core cell

9. The Contiguity Problem II 2. Enforce contiguity with a solution method that guarantees it (Gilbert et al., 1985; Diamond & Wright, 1991; Brookes, 1996). Implicit enumeration algorithm (Gilbert et al., 1985; Diamond & Wright, 1991). 1 0 1 0 1 0 Region-growing heuristic (Brookes, 1998).

10. The Shape Problem To date, each model has proposed an alternative approach for quantifying site shape. Wright et al. (1983)Perimeter Minor & Jacobs (1994)Perimeter / Area Gilbert et al. (1985)Perimeter * max. diameter Diamond & Wright (1991)(max. diameter squared) / area Brookes (1997)sinusoidal wave form shapes shape preference ordering best measure site search model best site Shape preference modeling

11. Problem Size Limitations The difficulty in solving existing site search models optimally is a function of the number of land units in the spatial data set as well as the number of expected spatial units in the site.

12. Review of Existing Approaches GIS Strengths Flexible approach to complex siting decisions. Computationally friendly. Weaknesses Doesn’t locate an explicit site. Doesn’t allow explicit site characteristics in a search. Spatial Optimization Strengths Locates an explicit site. Allows explicit site characteristics to be included in a search. Weaknesses The contiguity problem The shape problem Problem size limitations General Framework

13. Decomposing the Problem with a Site Search Neighborhood Operator How do we identify the feasible neighborhood around a root location? How do desired site shape and area affect the feasible neighborhood? solution root feasible neighborhood 1 0 s(x) 5 10 15 20 root feasible neighborhood

14. Site fields Spatially decomposes the problem into a set of smaller, local problems. Generates a larger number of alternatives. Serves as a conceptual construct amenable to a variety of model implementations.

15. 1-D Contiguity Constraints 0 1 1 2 3 4 5 6 7 8 9 r Suit. x y x must be included for y to be included

16. Shortest Path Contiguity (SPC) x z x or y must be included for z to be included y

17. Calculating the Feasible Neighborhood feasible neighborhood diameter root

18. Formulating a Site Search Model objectives constraints SPC constraints max

19. Conclusions A neighborhood operator provides a means to conceptually integrate the GIS and spatial optimization approaches to site search. The resulting methodological framework can be used to locate an explicit site optimally when spatial and aspatial characteristics of the site are included as either objectives or constraints. The approach decomposes optimal site search problems into a set of smaller, local problems. Thus, computational complexity is not an exponential function of the number of spatial units in the input data set.